Lipschitz conditions on bounded harmonic functions on the upper half-space

Abstract

This work is devoted to Lipschitz conditions on bounded harmonic functions on the upper half-space in ${\mathbb{R}}^n$. Among other results we prove the following one. Let $U(x^{\prime },x_n)$ be a real-valued bounded harmonic function on the upper half-space ${\mathbb{R}}_{+}^n=\{(x^{\prime },x_n):x^{\prime }\in {\mathbb{R}}^{n-1},x_n\in (0,\infty )\}$, which is continuous on the closure of this domain. Assume that for $\hat{I}\pm \in (0,1)$ there exists a constant $C$ such that for every $x^{\prime }\in {\mathbb{R}}^{n-1}$ we have $||U|(x^{\prime },x_n)-|U|(x^{\prime },0)|\le C{x}_n^{\hat{I}}\pm ,x_n\in (0,\infty )$. Then there exists a constant $\tilde{C}$ such that $|U(x)-U(y)|\le \tilde{C}|x-y{|}^{\hat{I}}\pm ,x,y\in {\mathbb{R}}_{+}^n$.